In Sections 2.5,
2.6, and 2.7,
evidence was presented indicating that
the velocity and spatial distribution of galaxies in regular, compact
clusters are in a relaxed,
quasistationary state. In this section, the nature of the relaxation
processes of galaxies in clusters will be briefly discussed.

If clusters result from the gravitational growth of initially small
perturbations, we expect the initial state of material in the
protocluster to be irregular. The relaxation of the cluster thus
involves the spatial motion of the galaxies, and a lower limit to the
relaxation time is the crossing time

(2.26)

where the radial velocity dispersion
r has been
substituted for the radial
velocity vr. An upper limit on the age of the cluster
is the Hubble time (age of the universe)

(2.27)

where the numerical coefficient depends on the cosmological model. Thus the
outer parts of a cluster or surrounding supercluster for which r 10 Mpc cannot
possibly have relaxed and are expected to be irregular, as is observed
(Sections 2.7 and
2.11.2).

The phase-space distribution of galaxies in the central parts of
spherical regular
clusters can be represented as f(r, v)
exp(- /
r2),
where 1/2 v2 +
(r) is the
energy per unit mass in the cluster
(Section 2.7). This is very similar
to a Maxwell-Boltzmann distribution, except that it is the energy per
unit mass (velocity dispersion) which is determined, and not the energy
(temperature). We first consider the possibility that clusters have relaxed
thermodynamically.

Thermodynamic equilibrium could result from elastic collisions between
galaxies, and would be expected if the time scale for energy transfer in
such
collisions were shorter than the age of the cluster or the time-scale
for the loss of
kinetic energy through dissipative processes. The most important elastic
two-body collisions in a cluster are gravitational. The resulting
relaxation times have been calculated analytically by
Chandrasekhar (1942) and
Spitzer and Harm (1958),
and numerically through N-body simulations, including a
realistic luminosity function for galaxies by
White (1976c,
1977b)
and Farouki and Salpeter
(1982).
A useful characteristic time-scale (0.94 times the Spitzer
'reference time') is

(2.28)

for the relaxation of a galaxy of mass m in a background field of
galaxies of mass mf and number density
nf.
is the ratio of
maximum to minimum impact
parameters of collisions which contribute to the relaxation; the maximum
impact
parameter is on the order of half the gravitation radius of the cluster
RG
(equation 2.21), while the minimum impact parameter is roughly the larger of
the galaxy radius rg or the 'turning radius'
Gmf /
3r2
(White, 1976b).
Thus

In order to give a lower limit to the relaxation time, we assume for the
moment that all the mass in a cluster (including the missing mass
(Section 2.8)) is bound to individual
galaxies; we later show this is
unlikely to be the case. Then, we define an average density
<>
3Mtot
/ (4RG3),
where Mtot is the total mass of the cluster. If we
assume a Schechter luminosity function (equation 2.3), and a fixed
galaxy mass-to-light ratio, we find

(2.30)

where m* is the characteristic galaxy mass
(corresponding to L*),
N* is the
characteristic galaxy number (richness), tcr is the
cluster crossing time (equation 2.26), and
(r) is
the total cluster density at r. Equation (2.30) is a
lower limit because it assumes all the cluster mass is bound to individual
galaxies. Typically,
RG
1 Mpc, r 1000 km/s,
N* 1000, and ln
3,
giving tcr(RG)
109 yr and
trel(m*,
RG)
3 × 1011 yr. This is much
longer than a Hubble time (equation 2.27); it is therefore unlikely that the
apparently relaxed state of regular clusters results from two-body
collisions. However, two-body relaxation processes can affect the more
massive galaxies (m >> m*) near
the cluster center
((r)
>> <>),
as is discussed below.